The Abstraction and Reasoning Corpus: D. Deferred Proofs

This paper is available on arxiv under CC 4.0 license. Authors: (1) Mattia Atzeni, EPFL, Switzerland and mattia.atzeni@outlook.it; (2) Mrinmaya Sachan, ETH Zurich, Switzerland; (3) Andreas Loukas, Prescient Design, Switzerland. Table of Links Abstract & Introduction Formalizing the Group-Action Learning Problem Attention Masks for Core Geometry Priors The LATFORMER Architecture Experiments Related Work Conclusion & References A. Additional Details on the Model B. Additional Experiments and Details on the Experimental Setup C. Limitations and Future Work D. Deferred Proofs D. Deferred Proofs We prove both Theorem 3.1 and 3.2 by induction on the dimensionality of the hypercubic lattice m. D.1. Base Case for Theorems 1 and 2 D.2. Inductive Step for Theorems 1 and 2 D.3. Proof of Corollary 1 The proof of Corollary 1 follows immediately from Theorem 3.2 and from the property of the Fourier transform according to which multiplying in the Fourier domain implements a convolution in the original domain. This paper is available on arxiv under CC 4.0 license. Authors: (1) Mattia Atzeni, EPFL, Switzerland and mattia.atzeni@outlook.it; (2) Mrinmaya Sachan, ETH Zurich, Switzerland; (3) Andreas Loukas, Prescient Design, Switzerland. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Mattia Atzeni, EPFL, Switzerland and mattia.atzeni@outlook.it; (2) Mrinmaya Sachan, ETH Zurich, Switzerland; (3) Andreas Loukas, Prescient Design, Switzerland. Table of Links Abstract & Introduction Formalizing the Group-Action Learning Problem Attention Masks for Core Geometry Priors The LATFORMER Architecture Experiments Related Work Conclusion & References A. Additional Details on the Model B. Additional Experiments and Details on the Experimental Setup C. Limitations and Future Work D. Deferred Proofs Abstract & Introduction Abstract & Introduction Formalizing the Group-Action Learning Problem Formalizing the Group-Action Learning Problem Attention Masks for Core Geometry Priors Attention Masks for Core Geometry Priors The LATFORMER Architecture The LATFORMER Architecture Experiments Experiments Related Work Related Work Conclusion & References Conclusion & References A. Additional Details on the Model A. Additional Details on the Model B. Additional Experiments and Details on the Experimental Setup B. Additional Experiments and Details on the Experimental Setup C. Limitations and Future Work C. Limitations and Future Work D. Deferred Proofs D. Deferred Proofs D. Deferred Proofs We prove both Theorem 3.1 and 3.2 by induction on the dimensionality of the hypercubic lattice m. D.1. Base Case for Theorems 1 and 2 D.2. Inductive Step for Theorems 1 and 2 D.3. Proof of Corollary 1 The proof of Corollary 1 follows immediately from Theorem 3.2 and from the property of the Fourier transform according to which multiplying in the Fourier domain implements a convolution in the original domain.